Denote by C(\(\sigma)\) (respectively by \(L_ p(\sigma)\), \(1\leq p\leq \infty)\) the space of continuous (respectively of p-integrable) functions on the unit sphere \(\sigma =\sigma^{n-1}\) of \(R^ n\). Let \(E_ N(f)_ C\) (respectively \(E_ N(f)_ p)\) be the best approximation of \(f\in C(\sigma)\) (respectively of \(f\in L_ p(\sigma))\) by spherical polynomials on \(\sigma\) of degree at most N. The authors consider the subspaces \('H^ r_{\infty}(\sigma)\subseteq C(\sigma)\) respectively \(H^ r_ p(\sigma)\subseteq L_ p(\sigma)\) endowed with the norms \(\| \cdot \|_{'H^ r_{\infty}(\sigma)}\) respectively \(\| \cdot \|_{H^ r_ p(\sigma)}\); their definitions are too technical to be reproduced here. Main results: Theorem 1.1. If \(f\in 'H^ r_{\infty}(\sigma)\) then \(E_ N(f)_ C\leq c\cdot M_ f/N^ r,\) \(N=1,2,...\), where c does not depend on N and f. Conversely, if \(E_ n(f)_ C\leq k/N^ r,\) \(N=1,2,...\), where k does not depend on N, then \(f\in 'H^ r_{\infty}(\sigma)\) and \(\| f\|_{'H^ r_{\infty}(\sigma)}\leq c_ 1(k+\| f\|_ C)\) where \(c_ 1\) does not depend on N and f. Theorem 1.2. If \(f\in H^ r_ p(\sigma)\) then \(E_ N(f)_ p\leq c\| f\|_{H^ r_ p(\sigma)}/N^ r,\) \(N=1,2,...\), where c does not depend on N and f. Conversely, if \(E_ N(f)_ p\leq k/N^ r,\) \(N=1,2,...\), then \(f\in H^ r_ p(\sigma)\) and \(\| f\|_{H^ r_ p(\sigma)}\leq c_ 1(\| f\|_ p+k).\)